Soccer teams in the in the world are gonna be meaning up in Russia this summer.

Not only do we obsess on the World Cup, but many of us also have this obsession about this panini World Cup album.

So this is the sticker album.

I'll show you the one from four years ago.

This is the one from World Cup 2014 in Brazil.

You get to see all the different countries that are participating, all the different players that are participating.

It's kind of sad.

I couldn't feel it like I was not able to get all the stickers.

But the idea is that hopefully you are able to get all of them, and you get to learn a lot of what the players on dhe just beam or engaged with it with what's happening.

So when you just buy, it would like I just did this morning.

It looks like this, and it's it's completely empty, and you can see, for example, Columbia.

This is being that I'm excited about.

And so these are the names of all the different players, and then you get these sticker backs.

So there's five in a five little stickers in a little pack.

So anybody who's been doing this for many years, we even know what this thing smells like.

It's like, you know, the smell of childhood right here.

You'll be excited about this one.

I just got massive gold won't go from Australia.

I take the stickers and stick them on the album.

I've been asking myself how much my paycheck am I gonna be spending on this because, you know, I'm a mathematician.

I could do a calculation, and I know that this is not a great idea.

Gonna be costing a lot of money, but I kind of want to think about how much is it gonna cost?

The sticker album this year has 682 stickers, the last one in 681.

But for some reason they decided that the 1st 1 should have number zero, So there's 682 stickers.

Each one of these packs costs a dollar.

There's five stickers in each one, and so that means that each sticker is 20 cents.

The difficult thing is that I don't get to choose what's what's in here.

So you know, if I got super lucky and I got all the stickers that I wanted with no repeats, then there would be 682 stickers when I multiply this.

This would be the cost of filling the album $136.40.

But of course, there's no way that I'm never gonna get that look.

Okay, I'm gonna you know, if anybody has done this before, they know that you keep getting repeats and repeats and repeats.

So what I'm asking myself is, how many stickers will I need to buy until I get all of Actually, this is something that people have tried to do before.

The most recent story that came out talked about how it would cost about $1000 to fill the album, and I see where they get that.

But I want I want to give us a different way of thinking about this, because the problem is that that estimate over $1000 assumes that you're just buying and buying and buying and buying stickers.

And that's not really how it works.

How it works is that many people around around you are buying stickers And so whenever you get repeats and they get repeats, then you're you're able to swap with each other.

And that way you're able to lower the costs.

Let's say they were trying to find his number and which is the number of stickers I expect to buy before I get every sticker at least once.

This is the lonely sticker collector version.

I mean, I'm I'm in the u.

S.

I have nobody to trade with.

And so I'm just going to imagine that I'm that I'm buying and buying stickers until I get Let me try to define And I Toby, how many stickers it took me to get between the Iast new sticker on that I plus one mistake.

Okay, So, for example, and zero will be how many stickers Until I get the first, No one and one will be how many doesn't get a second.

So you can imagine, for example, what is and 200.

That means that I've bought a lot of packs.

I have 200 the players, and now I really want to get the next one I'm looking for for some new one that I don't have in my album.

And then I keep I keep buying and buying and buying stickers until I get one that I don't have yet and however many stickers it takes me, I'm going to call that number and I the total number of stickers is going to be basically, you know, how long did it take me to get the first new one?

Plus, how long did it take me to get the second new one and so on until I get the last new one?

This is how long it's going to take me to find the very last.

And so if I want to find this number and how many stickers I need to buy, what I'm going to do is I'm going to actually try to find each one of these numbers.

And let's just say we were compute and 200 for an example.

What I want you to imagine is this.

I have 200 stickers, and now I'm going to buy a new one so you can imagine I open the back, I grab one of them.

And this is this is what I get Jonathan dos Santos from Mexico.

And what I do is that I have to look and see.

Do I already have Jonathan the fantasy?

Is he one of the 200 that I already have?

Or do I not having it?

And, of course, there's There's 682 possibilities off which from this could be this is number 461.

But, you know, it could be anything between 01 600 let's imagine that I put a circle around the ones that I already have.

So let's say that I already found sticker number one.

I haven't found zero I haven't found too.

Let's say that I did find 461 already, and I haven't found 600.

So pulled another sticker out.

Andi want you to imagine that this is undressed Christensen from Denmark number 257.

And now imagine that I opened my album and I find that undress Christensen is no one that I have already, so I can get to stick him right away.

And so it only took me one new sticker to get a one new player.

Okay, so if this was the 1st 1 that I drew, then it would take me one new sticker.

You know, if I happened to draw this one out, then it takes me one sticker to get a new one.

Here.

It takes me one.

Stick it to get a new one.

But now let's let's go back to the reality where I actually pull Jonathan Dos Santos And I did have Jonathan Jonathan Dos Santos already.

And so now the question is, how many stickers is it gonna take me from now on?

Well, this was a waste, you know.

I already had Jonathan Dos Santos.

He's not useful to me.

So I need to draw again.

How many new steps is going to take me to get the next one?

Well, this is precisely the definition of the number and 200.

So I wasted this step on days like I have to start over and it's still going to take me and 200 steps because that's what that number meets.

So if I draw Jonathan Doe centers, it's actually gonna take me one plus and 200 stickers before I get the same.

For example, if I draw number one, it's going to take me one place and 200 so basically depending on what I draw here, it's either gonna work right away.

I just draw one sticker on it's a new one, or if I already had it, then it's going to take me one plus and 200 stickers.

How many stickers did I already have?

I had 200 stickers already.

So in 200 of the draws, it's going to take me this many steps to get the next one.

Whereas for the stickers that I'm missing, which is a lot I'm missing 482 of them is going to go right away.

And so when I average these numbers But we're the total number which is 682 Well, that average is going to be how long it takes me to get that stick.

So I get this nice equation.

And now, if you look at this equation, you're gonna find a 200 plus 4 82 is equal to 6 82 So this equation is just saying this is equal to one plus 200 over 682 tens and 200.

What I get is that the number of stickers that I should expect to buy before I get the 200.

And first no one is going to be this.

So that's one computation.

Now, as you can imagine, the same argument will work for any one of these numbers.

Okay?

And so what I want to do now is General General Eyes and do this for any one of these numbers.

And I think the key thing that we should realize here is where did this 482 come from?

Well, you saw what happened.

It was 682 minus 200.

And I, which is how many stickers I need to buy between the ice new sticker and I place first new sticker is going to be 682 divided by 682 minus I.

So, if I combine these two, what I'm going to get is that in its equal to 682 overseas attorney to 680 to over 681 682 over 680.

You can see when you open your first pack, you're definitely going to get a sticky You didn't have exactly.

So this is, this is the number ends up zero.

How long is it gonna take me to get the first new sticker for sure, No matter what the 1st 1 I get is that's gonna just gonna come right away.

And as you go on, it gets harder and harder and harder and harder to get a new what is the last one.

So I get 692 divided by 6 32 minutes 6 81 which is equal to one that makes sense as well, because when you win this one sticking in aid, you've got a one in 682 chance of getting and it's so hard to get it.

And I think it's worth doubling on this point a little bit.

This is how long it takes you to get the very last sticker OK, and anybody who's collected this was collected.

Anything knows that this gets so much harder.

And at the very end, if I already got every single sticker and I'm only missing one, and I asked myself, How many should I buy?

Until I get that, no one is gonna take me a whole album's worth just to get their last vicar.

So I think at this point there's two kinds of people.

There's one kind of people who will be like, Oh, this is so beautiful store the mathematics of this and the people that will say What was the answer?

Let's talk about what the answer is first and then we'll come back to the beautiful mathematics behind this.

What is this number I compute?

Computed?

I went to a computer.

I worked it out that this number is about 4844.

So this is the number of stickers that I should expect to buy.

That's a lot of money.

I'm not about to do that.

I don't have that kind of paycheck.

What if you use the service that panini offers, which is that once you have 50 stickers that you're missing, then you can just order them from them?

Just send me those, and then I'll fill my album, and I don't have to trade with anybody.

I basically have to do this computation, except that instead of going all the way from zero 2 682 I want to do it from zero too clever and 32.

It's about 1775 stickers saving, so we're coming down from 1000 to 355.

And I think that's pretty shocking that to get the 1st 632 stickers cost you 355.

And to get the last 50 it's like over $600.

Do we have to pay a penny for those 50?

We do, and I have to be honest.

I think they do it at face value, but I'm not totally sure, but that's not fun.

I mean, the whole point of this is to buy stickers and traded with other people and so on.

And so I think we should go for a little bit of a more realistic scenario where we have a bunch of people that are trying to fill the album together.

Thankfully, there's this nice model that Donald Newman and Lawrence Shep came up with.

Let's say that we want to buy stickers among a bunch of people in this case and the 682 that's the number of stickers and F is just however many friends.

There are part of this game, so Federico, the model that you're about to talk to me about isn't how hard it will be to fill one album.

It's fact everyone to feel their album.

All the friends will feel their albums exactly.

So it's so it's It's a collective goal.

Everybody wants to fill the album.

We're all gonna help each other out.

And so they came up with this amazing formula.

I should warn you, that is kind of an ugly formula.

It's a big formula.

It's messy, and I'm not even expecting that everybody will understand it.

But the whole point that I want to make is that there is a former.

Okay, so let me show you that messy formula number of stickers divided by a number of friends so far, this is fine.

Now this is where it gets messy.

It's an integral from zero to infinity, off some very big expression.

Looks like this one minus big parenthesis, one minus.

It's a question here.

Here I get one plus X plus X squared to say, except for one X squared over one times two x cubed over one times, two times three and so on until the number of friends minus one take all of that and I divided by either the ex.

I warned you, This is messy.

Now you take a parenthesis here and you take this whole thing and raise it to the end, which is the number of stickers, and then you close parenthesis and then senator integral with variable X.

So integral here, Dex.

It's a monster.

It's a monster.

I think it's beautiful, but you have to have a taste for these kinds of things.

When I think it's beautiful, is not that I see this and I think it's really beautiful.

What I think is beautiful is that I just need to plug in and for the number of stickers f for the number of people.

And I'm going to get the answer for how many stickers per person we expect to be.

How much of a saving is it to be a sticker swapper?

So what we need to do is figure out howto plug in this thing.

One thing I will say is that if the number end and the number F are pretty small, then this expression is actually not too bad.

And if you've taken a course in integral calculus, you will have the tools to compute this for an f small.

But the problem is that we're looking at N is equal to 682.

We want to do this for a burial values of F.